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Extremal solutions of systems of measure differential equations and applications in the study of

Stieltjes differential problems

Rodrigo López Pouso

1

, Ignacio Márquez Albés

B1

and Giselle A. Monteiro

2,3

1Department of Statistics, Mathematical Analysis and Optimization, University of Santiago de Compostela, Santiago de Compostela, Spain

2Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

3Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovakia

Received 12 January 2018, appeared 13 June 2018 Communicated by Christian Pötzsche

Abstract. We use lower and upper solutions to investigate the existence of the greatest and the least solutions for quasimonotone systems of measure differential equations.

The established results are then used to study the solvability of Stieltjes differential equations; a recent unification of discrete, continuous and impulsive systems. The applicability of our results is illustrated in a simple model for bacteria population.

Keywords: measure differential equations, extremal solutions, lower solution, upper solution, Stieltjes derivatives.

2010 Mathematics Subject Classification: 34A12, 34A34, 34A36, 45G15, 26A24.

1 Introduction

The method of lower and upper solutions traces as far back as 1886, with Peano’s work [22]. Despite that, the existence of extremal solutions and their relation to lower and upper solutions continue to be the subject matter of many research papers on ordinary differential equations; for instance [3,4,11,15]. In recent years, special attention has been devoted to the question of solutions for discontinuous nonlinear differential equations; see e.g. [1,2,10,14]. In this regard, a few steps have been done towards the development of the corresponding theory of extremal solutions for measure differential equations [19].

Measure differential equations, as introduced in [6], are integral equations featuring the Kurzweil–Stieltjes integral. These equations are known to generalize other types of equations, such as classical differential equations, equations with impulses, or dynamic equations on time scales; see [7,20]. In this paper we are concerned with vectorial measure differential equations of the form

~y(t) =y~0+

Z t

t0

~f(s,~y(s))d~g(s), t ∈ I, (1.1)

BCorresponding author. Email: ignacio.marquez@usc.es

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where I = [t0,t0+L], ~y0Rn, ~f : I×RnRn, ~g : I → Rn is nondecreasing and left- continuous. The integral on the right-hand side is to be understood as an integration process

‘component-by-component’ in the Kurzweil–Stieltjes sense (see Section 3 for details). Accord- ingly, the equality (1.1) corresponds to a system of measure differential equations.

The objective of this paper is to establish the existence of the greatest and the least solutions for (1.1) in the presence of a pair of well-ordered lower and upper solutions. For our purposes, quasimonotonicity plays a key role. The main contribution of this paper lies on the fact that the nonlinear term ~f is assumed to satisfy weakened continuity hypotheses. Besides answering one of the questions posed in [19], our results provide an insight into the solvability of equations with functional arguments.

A secondary, but interesting, issue is the impact of our results in the theory of Stieltjes differential equations (also known asg-differential equations), [9]. As we will see, the integral equation (1.1) includes as a particular caseg-differential systems like

~x0g(t) =~f(t,~x(t)) forg-a.a. t∈ I, ~x(t0) = ~x0, (1.2) where~x0g stands for the derivative with respect to a nondecreasing and left-continuous func- tion g (see [17] for definition and properties of the derivative). These equations have gain in popularity as they offer an unified approach for investigating discrete and continuous problems. Impulsive differential systems, for example, can be rewritten as (1.2) by using an appropriate derivatorgwith jump discontinuities at the times when impulses are prescribed.

Benefiting from the relation between (1.1) and (1.2), we establish new theorems on extremal solutions for Stieltjes differential systems which somehow complement the study initiated in [16]. As an illustration of our results, we analyze the dynamics of a bacteria population using g-differential systems.

2 Preliminaries

In what follows we summarize some useful results concerning regulated functions. For details see e.g. [8,12].

Recall that a function is called regulated if the one sided-limits exist at all points of the domain. It is well-known that regulated functions are bounded and have at most countably many points of discontinuity. Given a regulated function f : [a,b] →Rn, we define f(a−) =

f(a), f(b+) = f(b)and

+f(t) = f(t+)− f(t), ∆f(t) = f(t)− f(t−), t∈[a,b],

where f(t+)and f(t−)stand for the right-hand limit and the left-hand limit, respectively.

As usual, G([a,b],Rn)denotes the space of all regulated functions f : [a,b] → Rn, and it is a Banach space when equipped with the supremum normkfk = supt∈[a,b]kf(t)k. In the case whenn=1, we will write simplyG([a,b]).

Theorem 2.1. The following two statements are equivalent:

(i) f ∈G([a,b]);

(ii) for everyε>0there exists a division D : a = α0 < α1 < · · · < αν(D) = b such that for every j∈ {1, . . . ,ν(D)}and s,t∈ (αj1,αj), we have|f(s)− f(t)|< ε.

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Remark 2.2. Given a function f ∈ G([a,b])and an arbitraryε >0, we will denote byDf the division whose existence is guaranteed by Theorem2.1.

Compactness in the space of regulated functions is connected with the following notion.

Definition 2.3. A set A ⊂ G([a,b]) is said to be equiregulated if for every ε > 0 and every t0 ∈[a,b]there existsδ >0 such that:

|x(t)−x(t0+)|< ε for all t0 <t<t0+δ, x ∈ A,

|x(t)−x(t0−)|< ε for all t0δ< t<t0, x ∈ A.

In the lines of Theorem2.1, we have the following characterization of equiregulated sets of functions.

Lemma 2.4. The following statements are equivalent:

(i) A⊂G([a,b])is equiregulated;

(ii) for everyε>0there exists a division D : a = α0 < α1 < · · · < αν(D) = b such that for every f ∈ A, j∈ {1, . . . ,ν(D)}and s,t ∈(αj1,αj),we have|f(s)−f(t)|<ε.

The next theorem is the analogue of the Arzelà–Ascoli theorem in the space of regulated functions.

Theorem 2.5. A subset Aof G([a,b])is relatively compact if and only if it is equiregulated and the set{f(t): f ∈ A}is bounded for each t ∈[a,b].

Remark 2.6. At this point it is worth mentioning one particular case in which the assumptions ensuring compactness are satisfied. Let A ⊂ G([a,b]) be such that there exist M > 0 and nondecreasing function h:[a,b]→Rsatisfying

|f(v)− f(u)| ≤h(v)−h(u) for f ∈ A, [u,v]⊆ [a,b],

and |f(a)| ≤ M for all f ∈ A. By Lemma 2.4 the set A is equiregulated, and obviously {f(t): f ∈ A}is bounded for eacht∈ I. Thus, in this case,Ais relatively compact.

The following result is concerned with pointwise supremum of regulated functions and, to the best of our knowledge, it is not available in the literature.

Proposition 2.7. LetAbe a relatively compact subset of G([a,b]). Then, the functionξ :[a,b]→R given by

ξ(t) =sup{f(t): f ∈ A}, t∈[a,b], is regulated.

Proof. From Theorem2.5, we know that{f(t): f ∈ A}is bounded for eacht ∈[a,b]; therefore, the functionξ is well defined. Moreover, sinceAis equiregulated, by Lemma2.4, givenε>0 there exists a division D : a = α0 < α1 < · · · < αν(D) = bsuch that for j∈ {1, . . . ,ν(D)}we have

|f(t)− f(s)|<ε for alls,t∈ (αj1,αj), f ∈ A. Fixed an arbitraryj∈ {1, . . . ,ν(D)}, fors,t∈ (αj1,αj)we get

f(s)−ε< f(t)< f(s) +εξ(s) +ε for all f ∈ A, and consequently

ξ(s)−εξ(t)≤ ξ(s) +ε.

In summary,ξ satisfies assumption (ii) of Theorem2.1, henceξ is regulated.

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In what follows, given functions f : [a,b]→ Rn and g : [a,b]→ R, the Kurzweil–Stieltjes integral of fwith respect togon[a,b]will be denoted byRb

a f(s)dg(s), or simplyRb

a fdg. Such an integral has the usual properties of linearity, additivity with respect to adjacent subinter- vals, as well as the properties to be presented next. The interested reader is referred to [23] or [26].

The following result is known as Hake property.

Theorem 2.8. Let f :[a,b]→Rnand g:[a,b]→Rbe given.

1. If the integralRb

t fdg exists for every t∈(a,b], and A∈Rnis such that

tlima+

Z b

t fdg+ f(a)(g(t)−g(a))

= A, thenRb

a fdg exists and equals A.

2. If the integralRt

a fdg exists for every t∈ [a,b), and A∈Rnis such that

tlimb

Z t

a fdg+ f(b)(g(b)−g(t))

= A, thenRb

a fdg exists and equals A.

Next theorem summarizes some properties of the indefinite Kurzweil–Stieltjes integral.

Theorem 2.9. Let f : [a,b] → Rnand g ∈ G([a,b])be such that the integral Rb

a fdg exists. Then the function

h(t) =

Z t

a fdg, t ∈[a,b], is regulated and satisfies

h(t+) = h(t) + f(t)+g(t), t∈[a,b), h(t−) =h(t)− f(t)g(t), t∈(a,b].

3 Vectorial measure differential equations

Measure differential equations, in the sense introduced in [6], are integral equations of the form

y(t) =y0+

Z t

t0

f(s,y(s))dg(s), t∈ I,

where the integral is understood as the Kurzweil–Stieltjes integral with respect to a nonde- creasing functiong: I →Rand the function f takes values inRn,n∈N. Herein we propose a more general version of such an equation where not only f can be a vectorial function but also the integratorg. More precisely, we are interested on equations

~y(t) =y~0+

Z t

t0

~f(s,~y(s))d~g(s), t∈ I, (3.1) where I = [t0,t0+L], ~y0Rn, ~f : I ×RnRn and ~g : I → Rn. The integral on the right-hand side is simply a notation for a ‘component-by-component’ integration process in the Kurzweil–Stieltjes sense, that is, writing

~y0= (y0,1, . . . ,y0,n), ~y= (y1, . . . ,yn), ~f = (f1, . . . ,fn), ~g= (g1, . . . ,gn),

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equation (3.1) corresponds to a systems ofnscalar equations, each of which reads as follows yi(t) =y0,i+

Z t

t0

fi(s,~y(s))dgi(s), t ∈ I, i∈ {1, . . . ,n}. (3.2) This interpretation of the integral in (3.1) is justified by the following vectorial equation

~y(t) =~y0+

Z t

t0

d[G(s)]~f(s,~y(s)), t ∈ I, (3.3) where for each t∈ I,G(t)∈Rn×nis the diagonal matrix

G(t) =

g1(t) 0 . . . 0 0 g2(t) . . . 0 ... ... . .. ... 0 0 . . . gn(t)

 .

In the case when ~f(t,~y(t)) =~y(t), equation (3.3) becomes a particular case of the so-called generalized linear differential equation; a branch of Kurzweil equations theory which has been extensively investigated in [23,26].

Clearly, by taking gi = g: I →Rfor alli∈ {1, . . . ,n}, from (3.1) we retrieve the notion of measure differential equation introduced in [6]. To avoid any misunderstanding, equations of the type (3.1) will be called vectorial measure differential equations. That said, we define the concept of solution for vectorial measure differential equations.

Definition 3.1. A function ~y ∈ G(I,Rn),~y = (y1, . . . ,yn), is a solution of equation (3.1) if yi : I →Rsatisfies (3.2) for eachi∈ {1, . . . ,n}.

In view of Theorem 2.9, we can see that a solution of (3.1) somehow shares the disconti- nuity points of~g.

Next we introduce the key concepts related to the question of extremal solutions. For that, we consider a partial ordering inRn as follows: given two vectors~x = (x1, . . . ,xn)and

~y= (y1, . . . ,yn), we write~x≤~yifxi ≤yi for eachi∈ {1, . . . ,n}. Naturally, for functions~α,~β: I →Rn, we write~α≤~βprovided~α(t)≤~β(t)for everyt ∈ I. Moreover, if~α,~β∈G(I,Rn)are such that~α≤~β, then we define the functional interval

[~α,~β] ={~η∈G(I,Rn) :~α≤~η≤~β}.

When we want to emphasize that we are considering (lower/upper) solutions which belong to a certain [~α,~β], we say that~ϕis a (lower/upper) solution between~αand~β.

The extremal (greatest and least) solutions to vectorial measure differential equations are defined in the standard way considering the aforementioned ordering, that is, if~y :I →Rnis a solution of (3.1) we say that:

•~yis the greatest solution of (3.1) on I if any other solution~x : I →Rnsatisfies~x ≤~y;

•~yis the least solution of (3.1) on I if any other solution~x: I →Rnsatisfies~y≤~x.

In the following sections we will investigate the existence of extremal solutions for (3.1) between lower and upper solutions.

Definition 3.2. A lower solution of (3.1) is a function~α∈G(I,Rn)such that~α(t0)≤ ~y0and αi(v)−αi(u)≤

Z v

u fi(s,~α(s))dgi(s), [u,v]⊆ I, i∈ {1, . . . ,n}.

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Symmetrically, an upper solution of (3.1) is a function~β∈ G(I,Rn)such that~y0≤~β(t0)and βi(v)−βi(u)≥

Z v

u fi(s,~β(s))dgi(s), [u,v]⊆ I, i∈ {1, . . . ,n}. Remark 3.3. If~α: I →Rnis a lower solution of (3.1), then for alli∈ {1, . . . ,n}

+αi(t) =αi(t+)−αi(t)≤ fi(t,~α(t))+gi(t), t∈ I, (3.4)

αi(t) =αi(t)−αi(t−)≤ fi(t,~α(t))gi(t), t∈ I. (3.5) Obviously, the reverse inequalities hold for upper solutions.

4 An existence result for the scalar case

In this section, we turn our attention to the scalar case of equation (3.1), that is, equations of the form

x(t) = x0+

Z t

t0

f(s,x(s))dg(s), t ∈ I, (4.1) wherex0R, f : I×RRandg: I →Ris nondecreasing and left-continuous.

The existence of the greatest and the least solutions for (4.1) has been already investigated in [19]. In this section, we address one of the questions posed in [19], namely, the existence of (extremal) solutions between given lower and upper solutions. Our result somehow general- izes what is available in the classical theory of ordinary differential equations (cf. [13]) as the function f is not required to be continuous with respect to the first variable.

For the convenience of the reader we will recall the main results in [19]. Given a setB⊆R, we consider the following conditions.

(C1) The integralRt0+L

t0 f(t,y)dg(t)exists for every y∈ B.

(C2) There exists a functionM: I →R, which is Kurzweil–Stieltjes integrable with respect to g, such that

Z v

u f(t,y)dg(t)

Z v

u M(t)dg(t) for everyy∈ Band[u,v]⊆ I.

(C3) For eacht∈ I, the mapping y7→ f(t,y)is continuous in B.

Next lemma is an important tool for dealing with conditions above (see [19, Lemma 3.1]).

Lemma 4.1. Assume that f : I×B→Rsatisfies conditions(C1), (C2), (C3). Then for each function x∈ G(I,B), the integralRt0+L

t0 f(t,x(t))dg(t)exists, and we have

Z v

u f(t,x(t))dg(t)

Z v

u M(t)dg(t), [u,v]⊆ I. (4.2) The following existence result for equations (4.1) derives from [19, Theorem 3.2].

Theorem 4.2. If f : I×RRsatisfies conditions(C1),(C2),(C3)with B=R, then equation(4.1) has a solution on I.

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Based on Theorems 4.4 and 4.12 from [19], we deduce the following result about extremal solutions.

Theorem 4.3. Suppose that f : I×RRsatisfies conditions(C1),(C2),(C3)with B=R. Further, assume that

(C4) for each t ∈ I the mapping u∈R7→u+ f(t,u)+g(t)is nondecreasing.

If equation(4.1)has a solution on I, then it has the greatest solution x and the least solution x on I.

Moreover, for each t∈ I we have

x(t) =sup{α(t): αis a lower solution of (4.1)on[t0,t]}, x(t) =inf{β(t): βis an upper solution of (4.1)on [t0,t]}.

Now, we will investigate the existence of extremal solutions for equation (4.1) provided a lower and an upper solutions are known and well-ordered.

Theorem 4.4. Suppose that(4.1)has a lower solutionαand an upper solutionβsuch thatα(t)≤ β(t) for all t ∈ I. Assume that the following conditions hold.

(H1) The integralRt0+L

t0 f(t,y)dg(t)exists for every y∈ E= [infsIα(s), supsIβ(s)].

(H2) There exists a function M : I → R, which is Kurzweil–Stieltjes integrable with respect to g,

such that

Z v

u f(t,y)dg(t)

Z v

u M(t)dg(t) for every y∈ E and[u,v]⊆ I.

(H3) For each t∈ I, the mapping y7→ f(t,y)is continuous in E.

(H4) For each t∈ I the mapping

u∈[α(t),β(t)]7→ u+ f(t,u)+g(t) is nondecreasing.

Then equation(4.1)has extremal solutions betweenαandβ. Moreover, for each t∈ I we have

x(t) =sup{`(t) : `lower solution of (4.1)betweenαandβ}, (4.3) x(t) =inf{u(t) : u upper solution of(4.1)betweenαandβ}. (4.4) Proof. Let us define fe: I×RRas

fe(t,x) =





f(t,α(t)) ifx <α(t),

f(t,x) ifα(t)≤x≤ β(t), f(t,β(t)) ifx >β(t),

and consider the modified problem x(t) =x0+

Z t

t0

ef(s,x(s))dg(s), t∈ I. (4.5)

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Clearly, (H3) ensures that fesatisfies (C3) withB=R. To show that ef satisfies both conditions (C1) and (C2) withB = R, let y ∈ R and put m(t) = max{min{y,β(t)},α(t)}, t ∈ I. Thus, m ∈ G(I,E) and ef(t,y) = f(t,m(t)) for every t ∈ I. Note that Lemma 4.1 and conditions (H1)–(H3) imply that Rt0+L

t0 f(t,m(t))dg(t) exists and (4.2) hold; in other words, for y ∈ R, the integralRt0+L

t0 ef(t,y)dg(t)exists and

Z v

u

fe(t,y)dg(t)

Z v

u M(t)dg(t), [u,v]⊆ I.

In summary, ef satisfies the conditions of Theorem4.2and we conclude that (4.5) has a solution defined on the whole ofI. The existence of the greatest solution x and the least solution x

of (4.5) is then a consequence of Theorem4.3and assumption (H4).

It only remains to show that if x : I → R is an arbitrary solution of equation (4.5), then α(t) ≤ x(t) ≤ β(t), t ∈ I, thus proving that x is a solution of (4.1) and the functions x and x are the intended extremal solutions betweenαandβ. Reasoning by contradiction, assume that there exists somet1 ∈(t0,t0+L]such that

α(t1)> x(t1). (4.6)

Let t2 = sup{t ∈ [t0,t1) : α(t) ≤ x(t)}. By the definition of supremum, either α(t2) ≤ x(t2) (which includes the caset2 =t0) or there exists a sequence of points uk ∈ (t0,t2),k∈N, such thatuk → t2 andα(uk)≤ x(uk)for eachk ∈ N. Therefore,α(t2−)≤ x(t2−). Using (3.5) and the fact thatgis left-continuous, we get∆α(t2)≤0, that is,

α(t2)≤α(t2−)≤ x(t2−) =x(t2).

Hence, we must haveα(t2)≤x(t2)and consequentlyt2 < t1. We will show that this leads to a contradiction with (4.6). First, observe that

α(t1)−x(t1) =α(t1)−α(t2) +α(t2)−x(t1)≤

Z t1

t2

f(s,α(s))dg(s) +α(t2)−x(t1). (4.7) The definition oft2 implies thatx(t)<α(t), t∈(t2,t1]; thus, by Theorem2.8we have

Z t1

t2

f(s,α(s))dg(s) = lim

σt2+ Z t1

σ

f(s,α(s))dg(s) + f(t2,α(t2))+g(t2)

= lim

σt2+ Z t1

σ

fe(s,x(s))dg(s) + f(t2,α(t2))+g(t2)

=

Z t1

t2

fe(s,x(s))dg(s)− ef(t2,x(t2))+g(t2) + f(t2,α(t2))+g(t2)

=x(t1)−x(t2)− ef(t2,x(t2))+g(t2) + f(t2,α(t2))+g(t2). Combining this equality with (4.7) we obtain

α(t1)−x(t1)≤α(t2) + f(t2,α(t2))+g(t2)−x(t2)− ef(t2,x(t2))+g(t2).

At this point we need to distinguish two cases regarding the value of ef. If α(t2) ≤ x(t2) ≤ β(t2), then ef(t2,x(t2)) = f(t2,x(t2)). Since condition (H4) implies that

α(t2) + f(t,α(t2))+g(t2)≤ x(t2) + f(t,x(t2))+g(t2),

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we conclude that α(t1)−x(t1) ≤ 0, a contradiction. In the case when x(t2) > β(t2), then ef(t2,x(t2)) = f(t2,β(t2)), which implies that

α(t1)−x(t1)≤α(t2) + f(t2,α(t2))+g(t2)−β(t2)− f(t2,β(t2))+g(t2).

The contradiction again follows from condition (H4), now taking into accountα(t2)≤ β(t2). The proof thatx≤ βon I is analogous and we omit it. Moreover, equalities (4.3) and (4.4) follow from Theorem4.3.

5 Extremal solutions for vectorial measure differential equations

Our goal is to extend the results from Section 4 to the vectorial equation

~y(t) =y~0+

Z t

t0

~f(s,~y(s))d~g(s), t ∈ I, (5.1) where y~0Rn, ~f : I×RnRn and~g : I →Rn. Herein, we assume that~g = (g1, . . . ,gn)is nondecreasing and left-continuous, that is, for each i∈ {1, . . . ,n}, the function gi : I →R is nondecreasing and left-continuous.

Like in the theory of multidimensional equations, the function ~f is required to be quasi- monotone. Recall that a function ~f is quasimonotone nondecreasing in a set E ⊆ I×Rn if given t ∈ I and vectors~x = (x1, . . . ,xn), ~y = (y1, . . . ,yn) such that (t,~x),(t,~y) ∈ E, the following holds:

~x≤~y with xi =yi for somei∈ {1, . . . ,n} =⇒ fi(t,~x)≤ fi(t,~y).

In what follows,~ei, i ∈ {1, . . . ,n}, denotes the vector in Rn whose i-th term is 1 and all others are zero.

Theorem 5.1. Suppose that(5.1) has a lower solution~α and an upper solution ~β such that~α ≤ ~β and assume that ~f is quasimonotone nondecreasing in E = {(t,~x) ∈ I×Rn : ~α(t) ≤ ~x ≤ ~β(t)}. Furthermore, assume that the following conditions hold.

(H1) The integralRt0+L

t0 fi(t,~η(t))dgi(t)exists for every~η∈ [~α,~β]and i∈ {1, . . . ,n}.

(H2) For each i ∈ {1, . . . ,n}, there exists a function Mi : I → R, which is Kurzweil–Stieltjes integrable with respect to gi, such that

Z v

u fi(t,~η(t))dgi(t)

Z v

u Mi(t)dgi(t) for every~η∈[~α,~β]and[u,v]⊆ I.

(H3) For each~η∈ [~α,~β], i∈ {1, . . . ,n}, and t ∈ I, the mapping

u∈[αi(t),βi(t)]7→ fi(t,~η(t) + (u−ηi(t))~ei) is continuous.

(H4) For each~η∈ [~α,~β], i∈ {1, . . . ,n}, and t ∈ I, the mapping

u∈ [αi(t),βi(t)]7→u+ fi(t,~η(t) + (u−ηi(t))~ei)+gi(t) is nondecreasing.

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Then equation(5.1) has extremal solutions in[~α,~β]. Moreover, for t ∈ I, the greatest solution~y = (y1, . . . ,yn)is given by

yi(t) =sup{`i(t) : (`1, . . . ,`n)lower solution of (5.1)in[~α,~β]}, (5.2) and the least solution~y = (y,1, . . . ,y,n)is given by

y,i(t) =inf{ui(t) : (u1, . . . ,un)upper solution of (5.1)in[~α,~β]}. (5.3) Proof. For i ∈ {1, . . . ,n} and t ∈ I, put hi(t) = Rt

t0 Mi(s)dgi(s) where Mi is the function in (H2). Hence each function hi : I → R, i ∈ {1, . . . ,n}, is nondecreasing and left-continuous.

Let~L = (L1, . . . ,Ln)be an arbitrary lower solution of (5.1) in[~α,~β]and consider the setA of functions~η∈[~α,~β],~η= (η1, . . . ,ηn), satisfying the following two conditions:

~η(t0)≤y~0 and ηi(v)−ηi(u)≤

Z v

u Mi(s)dgi(s) for [u,v]⊆ I, i∈ {1, . . . ,n}; (5.4) for each i∈ {1, . . . ,n} and each ε>0, Dηi ⊂DLi∪Dhi. (5.5) It is not hard to see that~L∈ A; moreover, every solution of (5.1) in[~α,~β]belongs toA(in such a case, condition (5.5) is a consequence of(H2)).

Define~ξ = (ξ1, . . . ,ξn)where, for eachi∈ {1, . . . ,n}, ξi : I →Ris the function given by ξi(t) =sup{ηi(t):~η∈ A and ~ηis a lower solution}, t∈ I. (5.6) Note that, for eacht ∈ I andi∈ {1, . . . ,n}, the set{ηi(t):~η∈ A} ⊂ [αi(t),βi(t)]. Therefore, the supremum ξi(t) is well-defined. Moreover, condition (5.5) ensures that {ηi : ~η ∈ A} is equiregulated (see Lemma2.4). Thus, Theorem2.5together with Proposition2.7implies that ξi is regulated for eachi∈ {1, . . . ,n}, and consequently~ξ ∈ G(I,Rn).

Claim 1. ξ~is the greatest solution of (5.1)in[~α,~β].

Fix an arbitraryi∈ {1, . . . ,n}and define the functionΦi : I×RRby Φi(t,x) = fi(t,ξ~(t) + (x−ξi(t))~ei), t ∈ I, x∈R.

Since~ξ ≤~β, the quasimonotonicity of ~f yields

Z v

u Φi(s,βi(s))dgi(s)≤

Z v

u fi(s,~β(s))dgi(s)≤ βi(v)−β(u), [u,v]⊆ I, which shows thatβi : I →Ris an upper solution of the scalar problem

x(t) =y0,i+

Z t

t0

Φi(s,x(s))dgi(s), t∈ I. (5.7) Using a similar argument, we can show that for any lower solution~ηof (5.1) such that~η∈ A, the functionηi : I →R is a lower solution of (5.7) between αi andβi. Noting that Φi satisfies the conditions of Theorem 4.4, it follows that (5.7) has the greatest solution xi : I → R betweenαi andβi, and by (4.3)ηi(t)≤ xi(t),t∈ I, for any~η∈ Alower solution of (5.1). Since the argument is valid for eachi ∈ {1, . . . ,n}, we construct a functionx~ = (x1, . . . ,xn), and obviouslyξ~ ≤ ~x. The quasimonotonicity of~f yields

xi(v)−xi(u) =

Z v

u Φi(s,xi(s))dgi(s)≤

Z v

u fi(s,x~(s))dgi(s), [u,v]⊆ I,

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for each i∈ {1, . . . ,n}, that is,x~ is a lower solution of (5.1) in [~α,~β], and

|xi(v)−xi(u)|=

Z v

u Φi(s,xi(s))dgi(s)

=

Z v

u fi(s,ξ(s) + (xi(s)−ξi(s))~ei)dgi(s)

Z v

u Mi(s)dgi(s) =hi(v)−hi(u)

for every[u,v]⊆ I andi∈ {1, . . . ,n}. This shows that~xsatisfies (5.4) and (5.5). Thusx~∈ A, and the definition ofξ~ implies x~≤ ~ξ. In summary,~ξ = ~x and

ξi(t) =y0,i+

Z t

t0 Φi(s,ξi(s))dgi(s) =y0,i+

Z t

t0 fi(s,ξ~(s))dgi(s), t ∈ I, i∈ {1, . . . ,n}. Therefore, ~ξ is a solution of (5.1), and, by (5.6) it is the greatest one in[~α,~β].

Claim 2. The greatest solution of (5.1)in[~α,~β],y~ =ξ~, satisfies(5.2).

The lower solution~L ∈ [~α,~β]was fixed arbitrarily, so y~ is greater than or equal to any lower solution in[~α,~β]. On the other hand,y~ is a lower solution itself and so (5.2) holds.

The proof of the existence of the least solutiony~ as well the validity of (5.3) is analogous.

6 Extremal solutions for vectorial measure differential equations with functional arguments

We will now consider the functional problem

~y(t) =y~0+

Z t

t0

~f(s,~y(s),~y)d~g(s), t∈ I, (6.1) where ~y0Rn, ~f : I ×Rn×G(I,Rn) → Rn and~g : I → Rn is nondecreasing and left- continuous. We recall that the integral on the right-hand side should be understood as a vectorial Kurzweil–Stieltjes in the sense presented in Section 3.

Equations (6.1) subjected to functional arguments represent a quite general object. It is not hard to see that functional differential equations of the form

~y0(t) =~f(t,~y(t),~y)

can be regarded as (6.1) provided the integral of ~f exists in some sense (in such a case gi corresponds to the identity function for each i). The class of problems covered by (6.1) also includes the so-called measure functional differential equations in the sense introduced in [6].

To see this it is enough to consider

~f(t,~y(t),~y) =~F(t,~yt) and ~g= (g, . . . ,g),

wherer>0,~F: I×G([−r, 0],Rn)→Rn,g: I →Ris nondecreasing and left-continuous, and for each t ∈ I the function~yt : [−r, 0]→ Rn denotes the history or memory of~y in [t−r,t], that is,~yt(θ) =~y(t+θ),θ ∈[−r, 0].

Unlike the work developed in previous sections, to investigate the extremal solutions for the problem (6.1) we will use a fixed-point approach; namely, the following result which is a consequence of [11, Theorem 1.2.2].

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Proposition 6.1. Let~α,~β∈G(I,Rn)be such that~α≤~βand let T :[~α,~β]→[~α,~β]be a nondecreasing map. Assume that for each i∈ {1, . . . ,n}there exists a nondecreasing map hi : I → Rsuch that for all~γ∈[~α,~β]and all[u,v]⊆ I the following inequality holds

|(T~γ)i(v)−(T~γ)i(u)| ≤hi(v)−hi(u). (6.2) Then T has the least fixed point~γ and the greatest fixed point~γ in[~α,~β]. Moreover,

~γ =min{~γ∈[~α,~β]: T~γ≤~γ}, ~γ=max{~γ∈[~α,~β]:~γ≤ T~γ}.

Proof. We will apply [11, Theorem 1.2.2] assuming X = Y = G(I,Rn) equipped with the supremum norm and the partial ordering defined in Section 3. Given a monotone sequence {~γk}k=0 in [~α,~β], it suffices to show that for each i ∈ {1, . . . ,n} the sequence {(T~γk)i}k=0 converges inG(I). By (6.2) and Remark2.6, for eachi∈ {1, . . . ,n},{(T~γk)i}k=0 is a relatively compact subset ofG(I), hence it contains a convergent subsequence. The result then follows from the monotonicity of the sequence{(T~γk)i}k=0.

Next result is the analogue of Theorem5.1for functional equations. Note that the notion of lower and upper solutions for equation (6.1) is an obvious extension of Definition3.2. Indeed,

~α∈ G(I,Rn)is a lower solution of (6.1) provided~α(t0)≤ ~y0and αi(v)−αi(u)≤

Z v

u fi(s,~α(s),~α)dgi(s), [u,v]⊆ I, i∈ {1, . . . ,n}, while the reverse inequalities are used to define upper solutions of (6.1).

Theorem 6.2. Suppose that(6.1)has a lower solution~αand an upper solution~βsuch that~α≤~β. For each~γ ∈ [~α,~β], denote by ~f~

γ : I×RnRn the function defined as ~f~

γ(t,~x) = ~f(t,~x,~γ). Assume that for each~γ ∈ [~α,~β], the function~f~

γ is quasimonotone nondecreasing in E = {(t,~x) ∈ I×Rn :

~α(t)≤~x≤~β(t)}. Furthermore, assume that the following conditions hold.

(H1) The integralRt0+L

t0 (f~γ)i(s,~η(t))dgi(s)exists for every~γ,~η∈ [~α,~β]and i∈ {1, . . . ,n}. (H2) For each i ∈ {1, . . . ,n}, there exists Mi : I → R,which is Kurzweil–Stieltjes integrable with

respect to gi, such that

Z v

u

(f~γ)i(t,~η(t))dgi(t)

Z v

u

Mi(t)dgi(t) for every~γ,~η∈[~α,~β]and[u,v]⊆ I.

(H3) For each~γ,~η∈[~α,~β], i∈ {1, ...,n},and t∈ I, the mapping

u∈ [αi(t),βi(t)]7→(f~γ)i(t,~η(t) + (u−ηi(t))~ei) is continuous.

(H4) For each~γ,~η∈[~α,~β], i∈ {1, ...,n},and t∈ I, the mapping

u∈ [αi(t),βi(t)]7→u+ (f~γ)i(t,~η(t) + (u−ηi(t))~ei)+gi(t) is nondecreasing.

(H5) For each t∈[t0,t0+L)and~x ∈Rn,the mapping~f(t,~x,·)is nondecreasing on[~α,~β]. Then equation(6.1)has extremal solutions in[~α,~β].

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Proof. Note that, for each~γ ∈ [~α,~β], assumption (H5)implies that~αand~β, respectively, are lower and upper solutions of the vectorial equation

~y(t) =y~0+

Z t

t0

~f~

γ(s,~y(s))d~g(s). (6.3) Consider the map T : [~α,~β] → [~α,~β] defined as follows: for each ~γ ∈ [~α,~β], T~γ is the greatest solution of (6.3) in[~α,~β]. The function T is well-defined as hypotheses (H1)–(H4) together with Theorem 5.1 guarantee the existence of extremal solutions of (6.3) in [~α,~β]. Moreover, T clearly satisfies (6.2) with hi(t) = Rt

t0Mi(s)dgi(s), i ∈ {1, . . . ,n}. In order to apply Proposition 6.1, we need to show that T is nondecreasing. Consider~γ,~η ∈ [~α,~β]such that ~γ ≤~η. By hypothesis (H5) we have ~f(s,T~η(s),~γ) ≤ ~f(s,T~η(s),~η) for s ∈ I. Thus, for i∈ {1, . . . ,n}and[u,v]⊆ I we get

Z v

u

(f~γ)i(s,T~η(s))dgi(s)≤

Z v

u

(f~η)i(s,T~η(s))dgi(s) = (T~η)i(v)−(T~η)i(u), that is,T~ηis an upper solution of

~z(t) =y~0+

Z t

t0

~f

~γ(s,~z(s))d~g(s). (6.4) Theorem 5.1 guarantees that (6.4) has the greatest solution between~α and T~η. Since T~γ is the greatest solution of (6.4) in[~α,~β]it follows that T~γ ≤ T~η. Hence, T is nondecreasing and Proposition6.1yields thatT has the greatest fixed point~γ with

~γ =max{~γ∈[~α,~β]:~γ≤ T~γ}.

Naturally,~γ is a solution of (6.1) in[~α,~β]. Moreover, it is not hard to see that if~γ ∈ [~α,~β]is any other solution of (6.1), then~γ ≤ T~γ. Therefore, by the definition of~γ, we conclude that

~γ is the greatest solution of (6.1).

To prove the existence of the least solution for equation (6.1) we proceed in a similar way but redefining the functionTso thatT~γcorresponds to the least solution of equation (6.3).

Using the theorem above we can establish the existence of extremal solutions for measure functional differential equations:

~y(t) =~y(t0) +

Z t

t0

~F(s,~ys)dg(s), t ∈ I,

~yt0 =~φ,

(6.5)

where I = [t0,t0+L], r >0,~φ∈ G([−r, 0],Rn),~F: I×G([−r, 0],Rn)→Rn and g: I → Ris nondecreasing and left-continuous.

Theorem 6.3. Let J = [t0−r,t0]∪I. Let~α,~β∈G(J,Rn)be such that~αt0 ≤~φ≤~β

t0 and

~α(v)−~α(u)≤

Z v

u

~F(s,~αs)dg(s), [u,v]⊆ I, Z v

u

~F(s,~αs)dg(s)≤~β(v)−~β(u), [u,v]⊆ I. Assume that~α≤~βand consider the functional interval

[~α,~β]J = {~η∈G(J,Rn) :~α≤~η≤~β}. Further, assume that the following conditions hold.

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(a) The integralRt0+L

t0 Fi(s,~ys)dg(s)exists for every~y ∈[~α,~β]J and i∈ {1, . . . ,n}.

(b) For each i ∈ {1, . . . ,n}, there exists Mi : I → R,which is Kurzweil–Stieltjes integrable with respect to g, such that

Z v

u Fi(s,~ys)dg(s)

Z v

u Mi(s)dg(s) for every~γ∈ [~α,~β]J and[u,v]⊆ I.

(c) For each t ∈ I, the mapping ϕ∈ P 7→ ~F(t,ϕ)is nondecreasing, where P ⊂ G([−r, 0],Rn)is the set P={~ys : ~y∈[~α,~β]J, s∈ I}.

Then equation(6.5)has extremal solutions in[~α,~β]J.

Note that assumptions(H3) and(H4)do not play a role in Theorem 6.3 as the function

~f(t,~y(t),~y) =~F(t,~yt)does not depend on~y(t).

By setting g(t) = t, equation (6.5) corresponds to the integral form of the retarded func- tional differential equation

y0(t) =F(t,yt), t∈ I, yt0 =φ. (6.6) Regarding scalar equations (6.6), the existence of solutions between well-ordered lower and upper solutions has been investigated in [24]. Therein, a monotone interactive method is applied in order to obtain the extremal solutions. Although [24] deals with lower/upper solutions which might be discontinuous, the function in the right-hand side,F, is assumed to satisfy the usual Carathéodory conditions. On one hand, in our Theorem6.3no continuity is required; however, the monotonicity condition (c) is admittedly stronger than the assumption (P5) stated at [24, Theorem 4].

7 Applications to Stieltjes differential equations

Stieltjes differential equations are differential systems in which the usual notion of derivative is replaced by a differentiation process with respect to a given monotone function. The basic theory for such equations has been established in [9,17]. In this work, we will consider vectorial Stieltjes differential equations of the form

~y0~g(t) =~f(t,~y(t)) for~g–a.a.t∈ I, ~y(t0) =~y0, (7.1) where I = [t0,t0+L], y~0Rn, ~f : I×RnRn and~g : I → Rn with~g = (g1, . . . ,gn)such that, for each i ∈ {1, . . . ,n}, gi : I → R is nondecreasing and left-continuous. The prob- lem described by (7.1) should be understood as the following system of Stieltjes differential equations:

(yi)0g

i(t) = fi(t,~y(t)) forgi–a.a. t∈ I, yi(t0) =y0,i, i∈ {1, . . . ,n}. (7.2) For a thorough study of the Stieltjes derivative which appears in (7.2) we refer to [9,17].

We remark that the equations studied in [9] are contained in (7.1), corresponding to the particular choice gi = g : I → R for all i ∈ {1, . . . ,n}. The Stieltjes equations in [9] were

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investigated in the space ACg(I)of functions absolutely continuous with respect tognonde- creasing and left-continuous. Recall that a functiony ∈ ACg(I)if for everyε >0 there exists δ>0 such that

m j=1

|y(bj)−y(aj)|< ε

for any family {(aj,bj)} of disjoint subintervals of I satisfying ∑mj=1(g(bj)−g(aj)) < δ. Ex- tending the notion of solution found in [9], we will look for solutions of the vectorial problem (7.1) in the space

AC~g(I) =ACg1(I)× · · · × ACgn(I),

where~g= (g1, . . . ,gn): I →Rn is a nondecreasing left-continuous function.

Definition 7.1. A solution of equation (7.1) is a function~y∈ AC~g(I)such that (7.2) holds.

As a consequence of the Fundamental Theorem of Calculus for the Lebesgue–Stieltjes integral, [17], we have the following lemma.

Lemma 7.2. Let I = [t0,t0+L],y~0Rn, ~f : I×RnRnand~g : I →Rn with~g= (g1, . . . ,gn) such that, for each i ∈ {1, . . . ,n}, gi : I →Ris nondecreasing and left-continuous.

If~y ∈ AC~g(I)is a solution of (7.1), then yi(t) =y0,i+

Z

[t0,t) fi(s,~y(s))dµgi for all t∈ I, i∈ {1, . . . ,n}, (7.3) where the integral stands for the Lebesgue–Stieltjes integral with respect to the Lebesgue–Stieltjes mea- sureµgi induced by gi.

Conversely, if ~y = (y1, . . . ,yn) : I → Rn satisfies (7.3), then ~y ∈ AC~g(I) and it solves the vectorial Stieltjes differential equation(7.1).

Using the lemma above and recalling the relation between Lebesgue–Stieltjes and Kurzweil–Stieltjes integrals, [21], one can show that a solution of (7.1) is also a solution of the vectorial measure differential equation (3.1).

In [18], it is shown that, under very general assumptions, the integral equation (4.1) is equivalent to

y0g(t) = f(t,y(t)) mg-a.e., y(t0) =y0, (7.4) where mg stands for the Thomson’s variational measure (see S0µg in [25]) induced by a function g : I → R. In the case when g is nondecreasing, as proved in [5], the variational measure mg corresponds to the Lebesgue–Stieltjes outer measure µg. Therefore, if E ⊂ I and mg(E) = 0, then µg(E) = 0 and, consequently, E is Lebesgue–Stieltjes measurable with µg(E) =0. Accordingly, a solution of (7.4) also satisfies equation

y0g(t) = f(t,y(t)) forg-a.a. t∈ I, y(t0) =y0,

wherey∈ ACg(I)if and only if f(·,y(·))is integrable on I with respect togin the Lebesgue–

Stieltjes sense. Therefore, along similar lines of the results in [18], we can draw a correspon- dence between the solutions of

yi(t) =y0,i+

Z t

t0

fi(s,~y(s))dgi(s), t ∈ I, i∈ {1, . . . ,n},

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and the solutions of (yi)0g

i(t) = fi(t,~y(t)) forgi-a.a. t ∈ I, yi(t0) =y0,i, i∈ {1, . . . ,n}.

Having all this in mind, based on results of previous sections, we can establish the exis- tence of extremal solutions for Stieltjes differential equations (7.1). Note that extremal solu- tions to (7.1) are defined in the obvious way in regard to Definition7.1. We now introduce the concepts of lower and upper solutions for this problem.

Definition 7.3. A lower solution of (7.1) is a function~α∈ AC~g(I)such that~α(t0)≤~y0and (αi)0g

i(t)≤ fi(t,~α(t)) forgi-a.a. t∈ I, i∈ {1, . . . ,n}. (7.5) Analogously,~β∈ AC~g(I)is an upper solution of (7.1) if~y0≤~β(t0)and

(βi)0gi(t)≥ fi(t,~β(t)) forgi-a.a. t ∈ I, i∈ {1, . . . ,n}.

Remark 7.4. Every lower solution of (7.1) is also a lower solution of (4.1). Indeed, given a lower solution~α of (7.1), since αi ∈ ACgi(I) for each i ∈ {1, . . . ,n}, by [9, Theorem 5.1], for every[u,v]⊆ I we have

αi(v) =αi(u) +

Z

[u,v)

(αi)0gi(s)dµgi,

where the integral stands for the Lebesgue–Stieltjes integral with respect to the Lebesgue–

Stieltjes measureµgi induced bygi. Therefore, (7.5) implies αi(v)−αi(u)≤

Z

[u,v)fi(s,~α(s))dµgi [u,v]⊆ I, i∈ {1, . . . ,n}.

Recall that Lebesgue–Stieltjes integrability implies Kurzweil–Stieltjes integrability, [21]. This, together with the fact that gi is left-continuous, ensures that the Lebesgue–Stieltjes integral on right-hand side coincides with the Kurzweil–Stieltjes integralRv

u fi(s,~α(s))dgi(s), see [21].

Since functions in the spaceACgi(I)have bounded variation ([9, Proposition 5.2]), we conclude that~αis a lower solution of the integral equation (4.1).

Similar arguments show that every upper solution of (7.1) is also an upper solution of (4.1).

In [16], extremal solutions for (7.1) have been studied in the scalar case. In order to apply the results of previous sections to investigate the solutions of the vectorial problem (7.1) we will need the following lemma which corresponds to a particular case of [18, Lemma 2.22].

Lemma 7.5. Let g :[a,b]→Rbe nondecreasing and left-continuous. If f :[a,b]→Ris null g-a.e., thenRt

a f(s)dg(s) =0for every t∈ [a,b].

The following result, obtained from Theorem5.1, ensures the existence of solution for the vectorial problem (7.1) in the presence of lower and upper solutions.

Theorem 7.6. Suppose that(7.1) has a lower solution~α and an upper solution~βsuch that~α ≤ ~β.

Assume that ~f is quasimonotone nondecreasing in E = {(t,~x) : ~α(t) ≤ ~x ≤ ~β(t)} and that the following conditions hold.

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